In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from one such basis to another is done through an orthogonal transformation.
The most familiar coordinate systems are the two-dimensional and three-dimensional Cartesian coordinate systems. Cartesian tensors may be used with any Euclidean space, or more technically, any finite-dimensional vector space over the field of real numbers that has an inner product.
Use of Cartesian tensors occurs in physics and engineering, such as with the Cauchy stress tensor and the moment of inertia tensor in rigid body dynamics. Sometimes general curvilinear coordinates are convenient, as in high-deformation continuum mechanics, or even necessary, as in general relativity. While orthonormal bases may be found for some such coordinate systems (e.g. tangent to spherical coordinates), Cartesian tensors may provide considerable simplification for applications in which rotations of rectilinear coordinate axes suffice. The transformation is a passive transformation, since the coordinates are changed and not the physical system.
Cartesian basis and related terminology
Vectors in three dimensions
In 3D Euclidean space, , the standard basis is , , . Each basis vector points along the x-, y-, and z-axes, and the vectors are all unit vectors (or normalized), so the basis is orthonormal.
Throughout, when referring to Cartesian coordinates in three dimensions, a right-handed system is assumed and this is much more common than a left-handed system in practice, see orientation (vector space) for details.
For Cartesian tensors of order 1, a Cartesian vector can be written algebraically as a linear combination of the basis vectors , , :
where the coordinates of the vector with respect to the Cartesian basis are denoted , , . It is common and helpful to display the basis vectors as column vectors
when we have a coordinate vector in a column vector representation:
A row vector representation is also legitimate, although in the context of general curvilinear coordinate systems the row and column vector representations are used separately for specific reasons – see Einstein notation and covariance and contravariance of vectors for why.
The term "component" of a vector is ambiguous: it could refer to:
a specific coordinate of the vector such as (a scalar), and similarly for and , or the coordinate scalar-multiplying the corresponding basis vector, in which case the "-component" of is (a vector), and similarly for and .
A more general notation is tensor index notation, which has the flexibility of numerical values rather than fixed coordinate labels. The Cartesian labels are replaced by tensor indices in the basis vectors , , and coordinates , , . In general, the notation , , refers to any basis, and , , refers to the corresponding coordinate system; although here they are restricted to the Cartesian system. Then:
It is standard to use the Einstein notation—the summation sign for summation over an index that is present exactly twice within a term may be suppressed for notational conciseness:
An advantage of the index notation over coordinate-specific notations is the independence of the dimension of the underlying vector space, i.e. the same expression on the right hand side takes the same form in higher dimensions (see below). Previously, the Cartesian labels x, y, z were just labels and not indices. (It is informal to say "i = x, y, z").
Second-order tensors in three dimensions
A dyadic tensor T is an order-2 tensor formed by the tensor product of two Cartesian vectors and , written . Analogous to vectors, it can be written as a linear combination of the tensor basis , , ..., (the right-hand side of each identity is only an abbreviation, nothing more):
Representing each basis tensor as a matrix:
then can be represented more systematically as a matrix:
See matrix multiplication for the notational correspondence between matrices and the dot and tensor products.
More generally, whether or not is a tensor product of two vectors, it is always a linear combination of the basis tensors with coordinates , , ..., :
while in terms of tensor indices:
and in matrix form:
Second-order tensors occur naturally in physics and engineering when physical quantities have directional dependence in the system, often in a "stimulus-response" way. This can be mathematically seen through one aspect of tensors – they are multilinear functions. A second-order tensor T which takes in a vector u of some magnitude and direction will return a vector v; of a different magnitude and in a different direction to u, in general. The notation used for functions in mathematical analysis leads us to write ,, used throughout while the same idea can be expressed in matrix and index notations, see Appendix C. (including the summation convention), respectively:
By "linear", if for two scalars and and vectors and , then in function and index notations:
and similarly for the matrix notation. The function, matrix, and index notations all mean the same thing. The matrix forms provide a clear display of the components, while the index form allows easier tensor-algebraic manipulation of the formulae in a compact manner. Both provide the physical interpretation of directions; vectors have one direction, while second-order tensors connect two directions together. One can associate a tensor index or coordinate label with a basis vector direction.
The use of second-order tensors are the minimum to describe changes in magnitudes and directions of vectors, as the dot product of two vectors is always a scalar, while the cross product of two vectors is always a pseudovector perpendicular to the plane defined by the vectors, so these products of vectors alone cannot obtain a new vector of any magnitude in any direction. (See also below for more on the dot and cross products). The tensor product of two vectors is a second-order tensor, although this has no obvious directional interpretation by itself.
The previous idea can be continued: if takes in two vectors and , it will return a scalar . In function notation we write , while in matrix and index notations (including the summation convention) respectively:
The tensor T is linear in both input vectors. When vectors and tensors are written without reference to components, and indices are not used, sometimes a dot ⋅ is placed where summations over indices (known as tensor contractions) are taken. For the above cases:
motivated by the dot product notation:
More generally, a tensor of order which takes in vectors (where is between and inclusive) will return a tensor of order , see for further generalizations and details. The concepts above also apply to pseudovectors in the same way as for vectors. The vectors and tensors themselves can vary within throughout space, in which case we have vector fields and tensor fields, and can also depend on time.
Following are some examples:
An applied or given... ...to a material or object of... ...results in... ...in the material or object, given by: unit vector Cauchy stress tensor a traction force angular velocity moment of inertia an angular momentum a rotational kinetic energy electric field electrical conductivity a current density flow polarizability (related to the permittivity and electric susceptibility ) an induced polarization field magnetic field magnetic permeability a magnetic field
For the electrical conduction example, the index and matrix notations would be:
while for the rotational kinetic energy :
See also constitutive equation for more specialized examples.
Vectors and tensors in dimensions
In -dimensional Euclidean space over the real numbers, , the standard basis is denoted , , , ... . Each basis vector points along the positive axis, with the basis being orthonormal. Component of is given by the Kronecker delta:
A vector in takes the form:
Similarly for the order-2 tensor above, for each vector a and b in :
or more generally:
Transformations of Cartesian vectors (any number of dimensions)
Meaning of "invariance" under coordinate transformations
The position vector in is a simple and common example of a vector, and can be represented in any coordinate system. Consider the case of rectangular coordinate systems with orthonormal bases only. It is possible to have a coordinate system with rectangular geometry if the basis vectors are all mutually perpendicular and not normalized, in which case the basis is orthogonal but not orthonormal. However, orthonormal bases are easier to manipulate and are often used in practice. The following results are true for orthonormal bases, not orthogonal ones.
In one rectangular coordinate system, as a contravector has coordinates and basis vectors , while as a covector it has coordinates and basis covectors , and we have:
In another rectangular coordinate system, as a contravector has coordinates and basis , while as a covector it has coordinates and basis , and we have:
Each new coordinate is a function of all the old ones, and vice versa for the inverse function:
and similarly each new basis vector is a function of all the old ones, and vice versa for the inverse function:
for all , .
A vector is invariant under any change of basis, so if coordinates transform according to a transformation matrix , the bases transform according to the matrix inverse , and conversely if the coordinates transform according to inverse , the bases transform according to the matrix . The difference between each of these transformations is shown conventionally through the indices as superscripts for contravariance and subscripts for covariance, and the coordinates and bases are linearly transformed according to the following rules:
Vector elements Contravariant transformation law Covariant transformation law Coordinates Basis Any vector
where represents the entries of the transformation matrix (row number is and column number is ) and denotes the entries of the inverse matrix of the matrix .
If is an orthogonal transformation (orthogonal matrix), the objects transforming by it are defined as Cartesian tensors. This geometrically has the interpretation that a rectangular coordinate system is mapped to another rectangular coordinate system, in which the norm of the vector is preserved (and distances are preserved).
The determinant of is , which corresponds to two types of orthogonal transformation: () for rotations and () for improper rotations (including reflections).
There are considerable algebraic simplifications, the matrix transpose is the inverse from the definition of an orthogonal transformation:
From the previous table, orthogonal transformations of covectors and contravectors are identical. There is no need to differ between raising and lowering indices, and in this context and applications to physics and engineering the indices are usually all subscripted to remove confusion for exponents. All indices will be lowered in the remainder of this article. One can determine the actual raised and lowered indices by considering which quantities are covectors or contravectors, and the relevant transformation rules.
Exactly the same transformation rules apply to any vector , not only the position vector. If its components do not transform according to the rules, is not a vector.
Despite the similarity between the expressions above, for the change of coordinates such as , and the action of a tensor on a vector like , is not a tensor, but is. In the change of coordinates, is a matrix, used to relate two rectangular coordinate systems with orthonormal bases together. For the tensor relating a vector to a vector, the vectors and tensors throughout the equation all belong to the same coordinate system and basis.
Derivatives and Jacobian matrix elements